9 questions from 2023
Let the quadratic curve passing through the point \( (-1, 0) \) and touching the line \( y = x \) at \( (1, 1) \) be \( y = f(x) \). Then the x-intercept of the normal to the curve at the point \( (\alpha, \alpha + 1) \) in the first quadrant is ______.
Let \( m_1 \) and \( m_2 \) be the slopes of the tangents drawn from the point \( P(4, 1) \) to the hyperbola \( H: \dfrac{y^2}{25} - \dfrac{x^2}{16} = 1 \). If \( Q \) is the point from which the tangents drawn to \( H \) have slopes \( |m_1| \) and \( |m_2| \) and they make positive intercepts \( \alpha \) and \( \beta \) on the \( x \)-axis, then \( \dfrac{(PQ)^2}{\alpha \beta} \) is equal to ______.
A straight line cuts off the intercepts \( OA = a \) and \( OB = b \) on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin \( O \) to this makes an angle of \( \frac{\pi}{6} \) with positive direction of y-axis and the area of \( \triangle OAB \) is \( \frac{98}{3}\sqrt{3} \), then \( a^{2}-b^{2} \) is equal to:
If \( P(h, k) \) be a point on the parabola \( x = 4y^{2} \), which is nearest to the point \( Q(0, 33) \), then the distance of \( P \) from the directrix of the parabola \( y^{2} = 4(x + y) \) is equal to:
Let the ellipse \( E:x^{2}+9y^{2}=9 \) intersect the positive \( x \)- and \( y \)-axes at the points \( A \) and \( B \) respectively. Let the major axis of \( E \) be a diameter of the circle \( C \). Let the line passing through \( A \) and \( B \) meet the circle \( C \) at the point \( P \). If the area of the triangle with vertices \( A \), \( P \) and the origin \( O \) is \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m-n \) is equal to:
The combined equation of the two lines \( ax+by+c=0 \) and \( a'x+b'y+c'=0 \) can be written as \( (ax+by+c)(a'x+b'y+c')=0 \). The equation of the angle bisectors of the lines represented by the equation \( 2x^{2}+xy-3y^{2}=0 \) is:
Let the system of linear equations \( x+y+kz=2 \), \( 2x+3y-z=1 \), \( 3x+4y+2z=k \) have infinitely many solutions. Then the system\( (k+1)x+(2k-1)y=7 \)\( (2k+1)x+(k+5)y=10 \)has:
Let \( y=x+2 \), \( 4y=3x+6 \) and\( 3y=4x+1 \) be three tangent lines to the circle \( (x-h)^2+(y-k)^2=r^2 \). Then \( h+k \) is equal to:
Let a common tangent to the curves \( y^{2}=4x \) and \( (x-4)^{2}+y^{2}=16 \) touch the curves at the points \( P \) and \( Q \). Then \( (PQ)^{2} \) is equal to ________.
